2008/09/30 by Jeffrey Schenker · 2 citations
Computer Science · Mathematics · Physics and Astronomy · #Matrix Theory and Algorithms #Random Matrices and Applications #Spectral Theory in Mathematical Physics #math-ph #math.MP #msc:15A52 #msc:82D30
paper · pdf · doi:10.1007/s00220-009-0798-0
published as Comm. Math. Phys. 290 (2009), 1065-1097 · 30 pages; corrected typos
arxiv created 2009/01/07 · openalex publication_date 2009/04/17 · arxiv updated 2010/06/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29
It is shown that certain ensembles of random matrices with entries that vanish outside a band around the diagonal satisfy a localization condition on the resolvent which guarantees that eigenvectors have strong overlap with a vanishing fraction of standard basis vectors, provided the band width W raised to a power μ remains smaller than the matrix size N. For a Gaussian band ensemble, with matrix elements given by i.i.d. centered Gaussians within a band of width W, the estimate μ≤ 8 holds.